SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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isolated-neutral wye load) there is no reason to use the full αβ0 transform and we can delete the ZS row without consequence. In that case (where we know that 0 , which means ZS components are not present) we can then make the simplifications in Equation (D.27) or (D.28). But, if we have a system where ZS components are or could be present, and we choose to ignore that ZS component (that is, we are only interested in the αβ components), we may use Equation (D.26), but we may not use Equation (D.27) or (D.28). The reason is that ( 0 ) was assumed in deriving those forms, thus they are not valid if 0 . As an example, the voltages in state S1 are given by Equation (D.39). x V x V x V AM DC BM DC CM DC (D.39) Using the αβ0 transform (Equation D.32) (where k 2/3), they are equivalent to Equation (D.40). 4/3 xV x 0 x0 1/3V DC DC (D.40) Using the Clarke transform (Equation D.26) with k 2/3, the αβ components are given by Equation (D.41), which of course match the αβ components in Equation (D.40). 4/3 x V x 0 DC (D.41) Now using Equation (D.27) with k 2/3, the αβ components are given by Equation (D.42), which is clearly incorrect. x VDC x 0 (D.42) The set in Equation (D.39) does not satisfy 0 (that is, it is known to contain a ZS component). This does not mean, however, that the α- and β- components have somehow been encoded the ZS component. It means that Equation (D.27) relies on 0 to account for the contributions of phase-B and phase-C in determining the α- component, and since that does not hold, it cannot properly reconstruct them. The practical consequence is that for the motor control aspects (current measurement, space vector regulation, and SVM inverter) the ZS component cannot exist, it is never mentioned, and 328
the Clarke transform is used. However, the SVM inverter physically produces voltages with a ZS component so to understand its operation the αβ0 transform is required. Still, the detailed aspects of the SVM inverter can be ignored from a high-level perspective if one does not care how the inverter works or why the output waveforms look as they do. Inverse Clarke Transform The inverse of the αβ0 transform is simply given by the matrix inverse, which is known to exist because the transform matrix is not singular (it takes three linearly dependent variables to three linearly dependent variables). But the inverse of the Clarke transform does not exist because it is not square. The simplest way to obtain it is to substitute in a symbolic variable in the place of the ZS coefficients in the matrix of the αβ0 transform, as in Equation (D.43). x 1 1/2 1/2 xA x 0 3/2 3/2 x B x 0 c c c x C (D.43) Upon taking its inverse (Equation D.44) it is clear that the ZS coefficients only multiple the 0 x and since the Clarke transform does not use x 0 , the column can be deleted. x 1 0 1/2 A cx 2 x B 1/2 3 /2 1/2c x 3 x 1/2 3 /2 1/2c x C 0 (D.44) Once again, the matrix in Equation (D.44) agrees with the understanding presented in the discussion of Figure D.11: the phase-variable components consist of the αβ projection (given by the first two columns of the matrix in Equation D.44) plus the ZS component. Of course, the same ZS component x 0 is added to each phase. Phase Interference Matrix It is clear that there is a relationship between the terminal voltages and the neutral voltage of an impedance load. Thus, a change in one terminal’s voltage will affect the neutral voltage, which will change the other phases’ line-neutral voltages. This effect is known as “phase interference” [62, p.93] and it does not seem to be treated well in the literature. In this report, the effect is represented using the matrix Q. The derivation follows simply from the results already presented and the final result is given by Equation (D.45). 329
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SENSORLESS FIELD ORIENTED CONTROL O
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Table of Contents List of Figures..
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CHAPTER 5 - Field Oriented Control
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List of Figures Figure 2.1 - Radial
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Figure 3.29 - Arbitrary space vecto
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Figure 4.44 - ZS components of some
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Figure C.13 - Single-turn full-pitc
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List of Symbols Symbol Meaning Unit
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Abbreviations ACIM AC induction mot
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Superscripts * commanded value ref
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CHAPTER 1 - Introduction Historical
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vector theory to model a machine, w
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publications and magazines, and Int
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elationship between SVM and other P
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CHAPTER 2 - Fundamentals of Electri
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Preliminaries In reading the litera
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Taxonomy of Motors There is any num
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characteristics (the DC load curren
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torque component, sturdily construc
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Figure 2.7 - Cross sections of some
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draw the line between the solenoida
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These texts generally attribute the
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N B A B Y x(t) (2.6) The induc
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extended to the rotational case lat
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present analysis). When this is tru
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Figure 2.16 - Components of total f
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As aforementioned a spatial analysi
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clear the meaning, but the reader i
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Figure 2.20 - Elementary brushless
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called the per-phase torque functio
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Expressing bEMF in terms of the bEM
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Equation (2.44) is the component of
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d R( r) ke( r) kt( r) sin( r) (2.
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General Electromechanical Models Th
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Figure 2.28 - Phase-variable simula
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Trapezoidal and Sinusoidal BPMS Mot
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Figure 2.29 - Back-EMF and drive cu
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Figure 2.31 - Back-EMF and drive cu
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3 KT Ke 2 (sinusoidal) K 2 K (tra
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control scheme must keep sinusoidal
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with misconceptions. The primary is
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Part I - Sinusoidal BPMS Motors wit
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grounded-neutral wye or delta conne
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N e f A ( ) i 2 N e f B ( ) i 2
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Figure 3.7 - Developed view at zero
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Figure 3.9 - Developed view showing
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Although Equations (3.6) or (3.7) a
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direction. The contributions always
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sinusoidal electrical variable will
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Figure 3.14 - Cross section of two
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3 N e (3.6): f ( , t) I p cos(
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In the previous chapter the per-pha
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peak value is represented as an upp
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Figure 3.21 - Time relationship bet
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another and thus could be placed in
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more widely understood theory (such
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other methods, although it has clos
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The SV as a Vector The first facet
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j0 j0 j0 aˆ 1 0 1 bˆ j e e e
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j j ee 1 cos( ) (3.45) 2 3 j t x
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Notice that the amplitude of the MM
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Figure 3.27 - Equivalent MMF produc
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Ne 1 jj f , t i e i * e 2 2
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distribution that is cosinusoidal i
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3 Ne j t f Ie p 2 2 . This
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j set θ to anything. Taking the re
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epresent any physically-distributed
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lumped-parameter model in this repo
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x kCx (3.75) abc The set of varia
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phase variable axes; three axes in
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The component MMFs act away from th
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common choice is to force the power
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As an example, find the SV that cor
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the same magnitude, the projection
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Figure 3.32 - Arbitrary SV referenc
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universal and the choice of convent
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stator’s perspective and at R fr
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The potential discrepancy (p.108) w
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x cos( r) sin( r) xd x sin(
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It must be emphasized that we have
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in relative time like an oscillogra
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Part III - SV Theory Applied to Sin
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In Equation (3.135) Z has elements
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to use SV theory. It may be helpful
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Figure 3.42 - Coupling between d- a
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R R d R R v Ri Li j Li dt s
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current does not influence the valu
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d v Ri L i e dt d v Ri L i e
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Figure 3.47 - Simulation diagram fo
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e je ), and (2) it was demonstrate
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Overview of Voltage Source Inverter
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In addition to measuring voltages w
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then there would be periods during
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Figure 4.9 - Gating and POLE voltag
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Figure 4.11 - Ideal voltage wavefor
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inverter is called sine-triangle co
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and it offers the fastest dynamic r
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Figure 4.18 - Fundamental gain and
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In the literature this is called th
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voltage will be below (above) the b
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Figure 4.24 - Instantaneous phase-A
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Figure 4.26 - Base SVs showing the
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Figure 4.27 - Transformed voltage w
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also correspond to six-step squarew
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was fed to the SVM inverter as a SV
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Figure 4.31 - Temporal limit of inv
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Figure 4.34 - SVM overmodulation re
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Checking sextant s 23 again for thi
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SVM Implementation The block diagra
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has been made of using THI in the S
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different triplen signal would be i
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Similarly, when the modulation inde
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Summary and Conclusion The two-leve
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CHAPTER 5 - Field Oriented Control
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torque production; this is called d
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Figure 5.3 - Torque control of sinu
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obviously a change in perspective.
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Figure 5.9 - Comparison of referenc
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variables to the stationary referen
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Figure 5.13 - Torque control of sin
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To analyze a salient machine one mu
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Figure 5.16 - Torque functions: (a)
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Figure 5.19 - Buried permanent magn
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Figure 5.21 - Flux weakening in the
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appears that this is very much rela
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Stationary and Synchronous Regulato
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stationary regulator had. For compl
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This coupling can be mitigated by m
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educes to two independent circuits,
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D, some of the 120° methods are ap
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d d v Ri Ls i R dt dt The r
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section a state-space model for the
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Figure 6.6 - Full state feedback (o
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State-Space Model of BPMS Motor A s
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Figure 6.9 - FOC block diagram. The
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loop). A second way to implement th
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knowing the rotor position from ini
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[11] E. Clarke, Circuit analysis of
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Control Systems, Linear Analysis [4
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[77] J.M.D. Murphy, F.G. Turnbull,
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Machine Modeling, Analysis [103] J.
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THI, SVM, SPWM [127] J.A. Houldswor
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Wisconsin-Madison. [Online]. Availa
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Angle Tracking [175] D. Morgan, “
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Appendix A - Elementary Electromagn
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L i (A.8) S SS S SR S SS SR (A.9
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First the self-flux-linkage of an i
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vAN R iA LM iA eA d v R i
Page 297 and 298: Figure B.3 - One-half of the flux p
Page 299 and 300: Equations (B.27) and (B.28) are for
Page 301 and 302: Fundamental Relationships Figure C.
Page 303 and 304: Figure C.3 - Sinusoidal winding den
Page 305 and 306: manufactured compared to the steppe
Page 307 and 308: phase winding of Figure C.4 will ha
Page 309 and 310: Distributed N Fp i 2 (C.5) Figu
Page 311 and 312: alanced machine only odd harmonics
Page 313 and 314: sinusoidal windings with a sinusoid
Page 315 and 316: 0, and since the direction of inte
Page 317 and 318: That the rotor-stator flux linkage
Page 319 and 320: Figure C.17 - Summary of rotor-stat
Page 321 and 322: Figure C.19 - Amplitudes of harmoni
Page 323 and 324: triangle with an amplitude whose nu
Page 325 and 326: Returning to Figure C.18 it is clea
Page 327 and 328: torque will describe the torque pro
Page 329 and 330: However, Equation (D.2) is incorrec
Page 331 and 332: the same order as the PS set (arran
Page 333 and 334: Implications of the ZS Component Th
Page 335 and 336: Figure D.6 - Neutral voltage of loa
Page 337 and 338: 3v 3e 3v v v v v v v 3v AM BM CM A
Page 339 and 340: components of source voltage sum to
Page 341 and 342: Equation (D.25) and instantaneous q
Page 343 and 344: If a ZS component could possibly be
Page 345 and 346: xA X1cos( t) X3cos(3 t) X5cos(5 t)
Page 347: Figure D.11 - 3-D base vectors of 1
Page 351 and 352: Appendix E - Park Transforms This a
Page 353 and 354: Appendix F - Useful Mathematical Re
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